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The first 60's songs that come to mind for me would be the following. They may or may not be my favorite songs by the artists listed, but they definitely are the songs from the 60's that I think of first....
Rolling Stones - Gimme Shelter
Beatles - Come Together
The Animals - House of the Rising Sun
Simon and Garfunkel - Scarborough Fair
The Mamas & The Papas - California Dreamin'
CCR - Midnight Special
Smokey Robinson/Miracles - Tracks of My Tears
Johnny Rivers - Baby, I Need Your Lovin'
Jimi Hendrix - Foxey Lady
Elvis - Suspicious Minds
Temptations - I Can't Get Next to You
Four Tops - Reach Out I'll Be There
Ray Charles - Unchain My Heart
Free - I'll Be Creepin' /-or-/ I'm a Mover (love both of these tracks)
Cream - Strange Brew
Aretha Franklin - Chain of Fools
Led Zeppelin - Heartbreaker / Livin' Lovin' Maid
Kinks - All Day and All of the Night
I tried to stick with just one song per artist (other than with Free), even though it'd be easy to list several songs for many of them.